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Where can I find the ab-initio derivation of the Lagrangian in classical mechanics? How did Lagrange arrive at $L=T-V$ ? Respected Experts kindly explain to me. What is the need to construct $L=T-V$ for a one dimensional mass spring system?

Qmechanic
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K Kulasekarapandian
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  • T-V is kinetic minus potential energy so why not take the expression for the kinetic energy (in terms of $\dot x$) and the expression in terms of potential energy (in terms of $x$) and just subtract? – ZeroTheHero Jun 29 '21 at 15:23
  • Actually no Lagrangian is a difference between the kinetic and potential energies. Lagrangian depends on unknown variables that we denote in the same way as the solutions, but they are not solutions yet. They are unknown functions to be "varied". After obtaining the right equations for those unknown variables, we must apply the initial conditions rather than use the positions at $t_1$ and $t_2$. If we manage to resolve the corresponding "initial value problem", certain combinations of variables are called (and are known functions of time) kinetic and potential energies. – Vladimir Kalitvianski Jun 29 '21 at 17:13
  • For a one dimensional mass spring system the Lagrangian is simple and instructive because it corresponds to so called oscillators so widely spread in QM and QFT. – Vladimir Kalitvianski Jun 29 '21 at 17:22
  • Possible duplicate: https://physics.stackexchange.com/q/78138/2451 – Qmechanic Jun 29 '21 at 18:16

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It's nice question. A simple answer to it is, it's made such that when we solve the Euler-Lagrange Equation it gives Newton's Equation of Motion.

But we use the fact that Principle of least action is more fundamental and can be extended further to other phenomenon such as Electromagnetism and General Relativity.

Dev
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  • It seems that I understand Prof. Vladimir Kalitvianski explanation. But it would be helpful for me if I can have a simple example from Prof.Vladimir Kalitvianski explanation. One Argentina Prof. wrote L=T-V as excess of kinetic energy over potential energy like excess of expenditure over income. A request to prof. Vladimir Kalitvianski explanation. Kindly suggest me a web site, book, or any material so that I can understand your explanation. If I do any thing wrong in this stack kindly forgive me professors and experts. – K Kulasekarapandian Jul 13 '21 at 10:14